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Thermoelectric properties of junctions between metal and models of strongly

correlated semiconductors

Massimo Rontani1, 2, ∗ and L. J. Sham1

1Department of Physics, University of California San Diego, La Jolla, California 92093-03192INFM National Research Center on nanoStructures and bioSystems at Surfaces (S3) and

Dipartimento di Fisica, Universita degli Studi di Modena e Reggio Emilia, 41100 Modena, Italy(Dated: November 15, 2018)

We study the thermopower of a junction between a metal and a strongly correlated semiconductor.Both in the electronic ferroelectric regime and in the Kondo insulator regime the thermoelectricfigures of merit, ZT , of these junctions are compared with that of the ordinary semiconductor. Byinserting at the interface one or two monolayers of atoms different from the bulk, with a suitablechoice of rare-earth elements very high values of ZT can be reached at low temperatures. Thepotential of the junction as a thermoelectric device is discussed.

PACS numbers: 72.10.Fk, 73.40.Cg, 73.40.Ns, 73.50.Lw

I. INTRODUCTION

New thermoelectric coolers and power generators areunder massive investigation.1 These devices are in generaleven more reliable than commercial heat-exchange refrig-erators but their efficiency, in the best cases, is muchlower. The quality of the material need in a thermo-electric device is defined by the dimensionless figure ofmerit ZT , where T is the absolute temperature, and Zis expressed in Eq. (1) in terms of transport coefficients.Currently, the highest value of ZT , which is ∼ 1 at roomtemperature, is found in Bi-Te alloys.2 New devices wouldbe competitive with traditional refrigerators if ZT wereabout 3 ∼ 4: the major lack of high-ZT materials is attemperatures below 300 K.In this paper3 we propose a junction of metal and

strongly correlated semiconductor as the basis for a possi-ble efficient low-temperature thermoelectric device. Thissystem embodies previous intuitions that were recog-nized fruitful4,5,6,7,8,9,10,11,12,13,14,15,16 in different materi-als/devices such as rare-earth compounds, superlattices,and metal/superconductor junctions. We now outlinethese concepts and briefly review the related literature.From the definition

Z =Q2σ

(κe + κl), (1)

where Q is the absolute thermopower (Seebeck coeffi-cient), σ the electrical conductivity, and κe and κl theelectronic and lattice part, respectively, of the thermalconductivity, it follows that an ideal thermoelectric ma-terial should have high thermopower, high electrical con-ductivity, and low thermal conductivity. Semiconductorsseemed to have the optimum collection of these proper-ties, in contrast with metals, which have high σ but lowQ, and insulators, which have high Q but low σ. How-ever, in the last thirty years no substantial enhancementof ZT beyond ∼ 1 was obtained.A breakthrough came with the synthesis of new mate-

rials such as filled skutterudites, which have nearly bound

rare-earth atoms, closed in an atomic cage, whose “rat-tling” under thermal excitation scatters phonons, thendramatically reducing κl.

4 More generally, heavy atomsin compounds help with lowering κl. Mahan and Sofo5

also showed that the best bulk band structure for high Qis one with a sharp singularity in the density of states veryclose to the Fermi energy. These results provide the firstidea in the search for the best thermoelectric, namelyto look at rare-earth compounds as major candidates.In fact, mixed valence metallic compounds (e.g. CePd3,YbAl3) show high values of Q, but at the present timeno useful value of ZT has been reported.17

The second idea is that the best thermoelectric musthave an energy gap. Because in ordinary semiconductorsthe optimum band gap is predicted to be about 10 kBT(kB being the Boltzmann constant),6 one is led to con-sider small-gap semiconductors for low-temperature ap-plications. If the chemical composition of semiconductorsincludes transition metals or rare earths, conduction andvalence bands are frequently strongly renormalized bycorrelation effects, forming a temperature-dependent gap(see Fig. 2): this is the case for mixed-valent semiconduc-tors, usually cubic, whose relevant electronic propertiesmay be modeled by a f -flat band and a broad conduc-tion band, with two electrons per unit cell.18 This classof materials consists of two subclasses: The first is theKondo insulator, characterized by very strong Coulombinteraction between electrons on the same rare-earth site,usually described by the slave boson solution of the An-derson lattice hamiltonian.19,20 Mao and Bedell predicteda high value of ZT for bulk Kondo insulators (the lowerthe dimension, the higher the value):7 however, some ex-perimental reports seem to exclude this possibility.1,21,22

The second, called the electronic ferroelectric (FE),23

consists of semiconductors with high dielectric constants,such as SmB6 and Sm2Se3, and it is modeled by theself-consistent mean-field (MF) solution of the Falicov-Kimball Hamiltonian.24 The ground state of the insu-lating phase is found to be a coherent condensate of d-electron/f -hole pairs, giving a net built-in macroscopic

2

polarization which breaks the crystal inversion symmetryand makes the material ferroelectric.23

Another useful observation is that ZT is reasoned toincrease in quantum-well superlattices, due to the mod-ification of the density of states.8,9,10 Moreover, super-lattices with large thermal impedance mismatch betweenlayers seem very efficient at reducing κl because interfacesscatter phonons very effectively.11 However, if the trans-port is parallel to the layers, the parasitic κl of the biggergap material25,26 or the tunnelling between conductionlayers26 can drastically decrease ZT . Besides, if intra-layer transport is diffusive and described by bulk param-eters, ZT for the whole superlattice cannot be higherthan the maximum value for the single constituents.27

In addition to the above literature, this work wasstimulated by some recent advances in thermoelectricapplications of junctions. Nahum and coworkers12

built an electronic microrefrigerator based on a metal-insulator-superconductor (NIS) junction. Subsequentexperimental28 and theoretical work13,29,30 confirmedthis new idea. Edwards and coworkers13 showed thattunneling through structures with sharp energy featuresin the density of states, like quantum dots and NIS junc-tions, can be used for cryogenic cooling. It seems then anatural extension to us to study the junction between ametal and a strongly correlated semiconductor.

There were recent proposals for devices such as semi-conductor/metal superlattices with transport perpendic-ular to interfaces. Mahan and Woods14 suggested a mul-tilayer geometry with the thickness of the metallic layersmaller than the electronic mean free path and the semi-conductor acting as a potential barrier (thermionic re-frigeration). Independently, Moyzhes and Nemchinsky15

proposed a similar configuration with the metallic layerthickness comparable with the energy relaxation length.We cite also Min and Rowe’s idea16 of using Fermi-gas /liquid interfaces. While the analysis of electronic trans-port of Ref. 14 and 15 is not applicable to our study,because of the sharp energy profile of the transmissioncoefficient across the junction which we examine, and ofthe nature of the correlated semiconductor ground state,this experimental geometry, with the strongly correlatedsemiconductor replacing the barrier layer, could be im-plemented, as we suggest at the end of Sec. IVC.

In sum, we build on two key ideas from the above lit-erature for increasing ZT . One is to utilize the sharpenergy features in the density of states of bulk materi-als as in strongly correlated semiconductors. The otheris to exploit the good thermoelectric characteristics of ajunction. In this paper, we combine these ideas in ex-ploring the thermopower behavior of a junction betweena metal and different classes of semiconductors. We de-scribe the gapped material on one side of the junctionas the solution of the Falicov-Kimball Hamiltonian24 indifferent regimes. In particular, we consider the case of:(i) Electronic Ferroelectric (FE), where, because of theCoulomb interaction between f -holes and d-electrons, theMF insulating ground state is a condensate of excitons;23

(ii) Narrow Band semiconductor (NB), characterized bythe d-f band hybridization: it would be a Kondo insu-lator had we take into account the f -f electron repul-sion; (iii) Broad band semiconductor (SC), for compar-ison. In these three cases, we solve the electronic mo-tion across the junction by means of a two-band modelanalogous to the Bogoliubov-de Gennes equations31 in afinite-difference form. In addition to the clean interface,we consider an “impurity” overlayer made either of rare-earth atoms, with relevant atomic orbitals of f -type, orof atoms with d-type orbitals, like transition metals. Inthe latter case, electrons can hop from these “d-impurity”sites to adjacent neighbor atoms, while in the former f -impurity case hopping is assumed negligible. This sce-nario is motivated by recent advances in atomic layerfabrication. We compute ZT for the interface via a lin-ear response. We find that ZT can be greatly enhancedby the presence of a suitable f -impurity layer at the in-terface. In particular, for a fixed working temperature ofthe junction, an optimum energy of the f -impurity levelexists which maximizes ZT , especially at low tempera-tures. In these regimes, bulk thermal conductivity wouldbe dominated by phonons which would reduce ZT . How-ever, one can fabricate the junction with two materialswith large thermal impedance mismatch, so that phononscatterings at the interface decrease the thermal conduc-tivity. Thus, phonon conductivity would not diminish thehigh ZT found. To this aim we propose an experimen-tal setup, namely perpendicular transport in metal/FEsuperlattice.The structure of the paper is as follows: in Section

II we describe the model Hamiltonian, in Sec. III A wesolve the electronic motion across the junction, and inSec. III B we compute transport coefficients and ZT . InSec. IV we present and discuss our results, for the cleaninterface (IVA), and for the d- (IVB) and f - (IVC) im-purity layer. Also we briefly discuss some structural re-laxation effects of the interface (sec. IVD). Our conclu-sions are in Sec. V.

II. THE MODEL

We introduce the one-dimensional spinless Hamilto-nian H to model the motion across the junction alongthe z direction perpendicular to the interface between ametal and different types of semiconductor. H is givenby the sum of three terms:

H = Hmetal +Hinterface +HFK. (2)

Hmetal is a tight-binding Hamiltonian describing themetal on the left side of the junction:

Hmetal = (ε′d + eV )∑

j<0

d†jdj − t′∑

j<0

d†jdj+1 + H.c. (3)

Here dj destroys an electron of charge e < 0 at the lat-tice site with energy ε′d and position z = aj, a is the

3

+++ -

df

df df

~~

V

t t_ _

_

0

dfL dfR

dfL dfR

z

V_

-a

~ ~

a/2 a

d-sitef-site

impurities

��������

��������

��������

��������

��������

��������

��������

��������

UU U

V V

FIG. 1: Pictorial representation of the junction along the z-axis perpendicular to the interface: the atomic sites at z = ajhave d-type orbitals while those at z = a(j+1/2) have f -typeorbitals, even and odd under spatial inversion, respectively.Different symbols for the impurities at z = 0 and z = a/2 aredrawn; also the parameters of the Hamiltonian of Eq. (2) aredepicted.

lattice constant, t′ is the hopping parameter for nearest-neighbor sites, and V is the electrostatic potential acrossthe junction by applying an external bias. In our model,V is constant in the two bulk regions with a discontin-uous step at the interface, although the actual profileshould be determined self-consistently together with theelectronic density.The third term HFK in Eq. (2) is the Falicov-Kimball

Hamiltonian24 referring to the insulator on the right sideof the interface. In addition to the Coulomb interac-tion Udf between the d-electrons at sites z = aj andthe f -electrons localized on atoms at z = a(j + 1/2),we add a hybridization term (Vdf is the modulus of thehybridization integral) between d and f orbitals. Be-cause the crystal has inversion symmetry, this term hasto be odd, namely the hybridization term between sitesat z = ja and z = (j + 1/2)a and that between sites atz = (j + 1/2)a and z = (j + 1) a have opposit sign (seeFig. 1):

HFK = H(1)FK +H(2)

FK, (4)

H(1)FK = εd

∑

j>0

d†jdj + εf∑

j>0

f †j+1/2fj+1/2 − t∑

j≥0

d†jdj+1

− Vdf

∑

j>0

f †j+1/2dj + Vdf

∑

j>1

f †j−1/2dj + H.c., (5)

H(2)FK = Udf

∑

j>0

d†jdjf†

j+1/2fj+1/2

+ Udf

∑

j>1

d†jdjf†

j−1/2fj−1/2, (6)

where t is the hopping coefficient, and εf and εd are thef - and d-site energies, respectively.The second term Hinterface in Eq. (2) is the Hamilto-

nian at the interface, which describes the overlayer madeof d-sites at z = 0 and f -sites at z = a/2:

Hinterface = εd0d†0d0 + εf1/2f

†

1/2f1/2

+ UdfLd†0d0f

†

1/2f1/2 + UdfRd†1d1f

†

1/2f1/2

−VdfLf†

1/2d0 + VdfRf†

1/2d1 + H.c. (7)

Here we have included the possibility of “impurity”atoms at the interface, namely one with average energyεd0 at position z = 0 and another one with energy εf1/2at position z = a/2. Since the impurity atoms have or-bitals differing from those in the bulk, the associated Udf

and Vdf parameters generally change. We denote them

by UdfL, VdfL, UdfR, and VdfR, referring to the couplesof sites z = 0, a/2 and z = a/2, a, respectively (Fig. 1).We give the MF solution of the Hamiltonian H of

Eq. (2), by means of an approach (see Sec. III A) anal-ogous to the Bogoliubov-de Gennes method31 (BdG). Incontrast to the electron-electron pairing in the Bardeen-Cooper-Schrieffer (BCS) MF theory of superconductiv-ity, the pairing here occurs between d-electrons and f -holes. To proceed, we assume that

⟨

d†jdj

⟩

= nd ∀j ≥ 1, (8)

⟨

f †j+1/2fj+1/2

⟩

= nf ∀j ≥ 1, (9)

Udf

⟨

d†jfj+1/2

⟩

= ∆ ∀j ≥ 1,

Udf

⟨

d†jfj−1/2

⟩

= ∆ ∀j ≥ 2, (10)

⟨

d†0d0

⟩

= nd0,⟨

f †1/2f1/2

⟩

= nf1/2, (11)

UdfL

⟨

d†0f1/2

⟩

= ∆L, (12)

UdfR

⟨

d†1f1/2

⟩

= ∆R. (13)

Here 〈. . .〉 is the symbol for the quantum statistical av-erage. Note that, in addition to the usual mean or-bital occupations nd and nf of standard Hartree-Focktheory (0 ≤ nd, nf ≤ 1), we also introduce the non-vanishing pairing potential ∆, characteristic built-in co-herence of the d-electron/f -hole condensate. Follow-ing de Gennes,31 we now write an effective HamiltonianHMF, to be computed self-consistently together with theenergy spectrum:

HMF = Hmetal +HMFinterface +HMF

FK , (14)

4

HMFinterface = εd0d

†0d0 + εf1/2f

†

1/2f1/2 − ∆Lf†

1/2d0

− ∆Rf†

1/2d1 − VdfLf†

1/2d0 + VdfRf†

1/2d1 + H.c.

−UdfLnd0nf1/2 +∣

∣

∣∆L

∣

∣

∣

2

+∣

∣

∣∆R

∣

∣

∣

2

, (15)

HMFbulk = εd1d

†1d1 + εd

∑

j>1

d†jdj + εf∑

j>0

f †j+1/2fj+1/2

− t∑

j≥0

d†jdj+1 − Vdf

∑

j>0

f †j+1/2dj

+ Vdf

∑

j>1

f †j−1/2dj −∆∑

j>0

f †j+1/2dj

− ∆∑

j>1

f †j−1/2dj + H.c.− UdfRndnf1/2

− Udfndnf −∑

j>1

2Udfndnf +∑

j>0

2 |∆|2 . (16)

Here we have defined the renormalized energies

εd0 = εd0 + UdfL nf1/2,

εf1/2 = εf1/2 + UdfL nd0 + UdfR nd,

εd1 = εd + UdfR nf1/2 + Udf nf ,

εf = εf + 2Udf nd,

εd = εd + 2Udf nf . (17)

We shall treat the renormalized quantities εd0, εf1/2, εd1,εf , and εd as material parameters. For simplicity, weshall assume t′ > t, εd1 = εd, and ε′d = εd = 0, that isthe middle of the d-band on both sides of the junction isaligned, but the metal bandwidth (4t′) is larger than thesemiconductor bandwidth (4t). We will consider onlythe case εf = 0, that is the flat band is in the middleof the d-band in the right-side material (see Fig. 2).32

With these assumptions, and dropping all the constantterms in Eqs. (15-16), we obtain the expression of theHamiltonian HMF which we will use in the following:

HMF = eV∑

j<0

d†jdj + εd0d†0d0 + εf1/2f

†

1/2f1/2

− t′∑

j<0

d†jdj+1 − t∑

j≥0

d†jdj+1

−(∆L + VdfL)f†

1/2d0 − (∆R − VdfR)f†

1/2d1

−(∆ + Vdf )∑

j>0

f †j+1/2dj

−(∆− Vdf )∑

j>1

f †j−1/2dj + H.c. (18)

Note the different parity of ∆- and Vdf -terms: while theVdf -term is odd, the ∆-term is even.20

The values of averages on the left side of equations(8-13) are spatially inhomogeneous and should be deter-mined self-consistently and simultaneously with the solu-tion of the BdG equations. We make the approximation

−π/2a0 π/2a−3

−2

−1

0

1

2

3

ener

gy ε

(t u

nits

)

−π/2a0 π/2a −π/2a0 π/2a −π/2a0 π/2a

q−vector k−vector k−vector k−vector

LEFTSIDE

METALFE NB SC

(i) (ii) (iii)

µ

d−band

f−band

FIG. 2: Quasi-particle bulk energy bands of materials onboth sides of the junction vs wave vector at T = 0, for thethree cases under study, in the “semiconductor representa-tion.” All energies are in units of t, and ǫ = 0 corresponds tothe chemical potential µ. The second panel [case (i)] showsalso the “bare” band structure (dashed lines), i.e. that notrenormalized by correlation, with Vdf = ∆ = 0. Since thealignment of µ with respect to the left-hand side metal banddepends on the kind of bulk semiconductor on the right-handside, here we assume a metal/FE setup, with metal (semicon-ductor) bare bandwidth 4t′ (4t), and t′ = 5t. Note that in thecase (i) [electronic ferroelectric] and (ii) [narrow band semi-conductor] the gap is indirect, while it is direct in the case (iii)[broad band semiconductor]. Besides, in case (i) [(ii)] the bot-tom (top) of conduction (valence) band is much flatter thanthe top (bottom) of valence (conduction) band, while in case(iii) the curvature of the two bands close to the direct gap iscomparable. The trasformation k → π/a − k, ǫ → −ǫ mapsthe FE electron (hole) band into the NB hole (electron) band.In this plot parameters are: (i) ∆(T = 0)=0.49t, Vdf=0. (ii)∆=0, Vdf=0.49t. (iii) t′′=0.141t, Ve=0.071t.

that those are the bulk averages appropriate to each sideof the interface layer. In particular, we take the orderparameter ∆ as constant on the right side of the inter-face, and zero on the left side.33 The quantities ∆L, ∆R

are used to characterize the effect of the interface on ∆.

The temperature dependence of ∆ is much strongerthan that of the average occupation of orbitals. We de-termine numerically ∆, together with the chemical po-tential µ, as the bulk value deep inside the right materialin a self-consistent way. For this purpose we use the bulkHamiltonian HMF

bulk:

HMFbulk = −t

∑

j

d†jdj+1 − (∆ + Vdf )∑

j

f †j+1/2dj

− (∆− Vdf )∑

j

f †j−1/2dj + H.c. (19)

Here the index j runs over the whole space. We alwaysassume an electronic occupation of one electron per unit

5

cell (a cell contains one d-site and one f -site), namely

1

Ns

∑

j

⟨

d†jdj + f †j+1/2fj+1/2

⟩

= 1, (20)

where Ns is the number of cells. Eq. (20) allows us to cal-culate the chemical potential µ for the bulk. Because theeffective Hamiltonian of Eq. (19) is strongly temperature-dependent, so µ is. Since µ refers to the bulk value insidethe semiconductor, the metal on the left side acts as anelectron reservoir.We will consider three specific cases for the actual val-

ues of parameters in the Hamiltonian HMF of Eq. (18):

(i) Vdf → 0 and non-zero Udf (VdfL = VdfR = 0),i.e. the case FE of the electronic ferroelectric with neg-ligible hybridization.34 (ii) Udf → 0 and non-zero Vdf

(UdfL = UdfR = 0), i.e. the case NB of a f -band hy-bridized with a d-band, that is a narrow band semicon-ductor which would be a Kondo insulator if it were drivenby the condensation of slave bosons. (iii) A ordinary,“broad band” semiconductor SC, without center of in-version. In the latter case we still use HMF but we re-gard it simply as a one particle Hamiltonian where the“d” and “f” indices are pure labels, and we rename Vdf

as t, considering it as the odd part of a hopping coeffi-cient between nearest neighbor sites, ∆ as Ve, the evenpart, and t as t′′, a second nearest neighbor hopping co-efficient. We make the choice t ≫ Ve, Ve ∼ t′′, leading tobroad conduction and valence bands with different effec-tive masses.While in case (i) ∆ has to be determined self-

consistently and in case (ii) Vdf is a material parameter,there is a formal canonical transformation connecting (i)to (ii),

dj → d†je−iπj, fj+1/2 → f †j+1/2e

−iπ(j+1),

∆ → V ∗df , εd0 → −εd0,

εf1/2 → −εf1/2, eV → −eV,

∆L → V ∗dfL, ∆R → V ∗dfR, (21)

leading to the mapping:

HMF(Vdf = 0) → HMF(∆ = 0)

−∑

j<0

eV − εd0 − εf1/2. (22)

This transformation, in particular, maps the FE electron(hole) excitations into the NB hole (electron) excitations.

III. TRANSPORT ACROSS THE JUNCTION

In the first subsection, we solve the BdG equations; inthe second one, we compute the transport coefficients, inthe limit of small electric or thermal gradient across thejunction.

A. The Bogoliubov-de Gennes equations

We now find the canonical transformation that diag-onalizes HMF in a self-consistent way, i.e., we solve theBdG equations. In the present context, the BdG ap-proach is essentialy the Hartree-Fock treatment of thetwo-band model. It is convenient to refer all the excita-tion energies to the chemical potential µ. Thus, we definethe number operator N ,

N =∑

j

(

d†jdj + f †j+1/2fj+1/2

)

,

and we replace the HamiltonianHMF withHMF−µN . Todiagonalize it, we introduce the unitary transformation

γke =1√Ns

∑

j

[u∗k(j) dj + v∗k(j + 1/2) fj+1/2],

γ†−kh =1√Ns

∑

j

[u−k(j) dj

+ v−k(j + 1/2) fj+1/2], (23)

where k is simply a quantum index, equivalent to thecrystal momentum only in the bulk case. The idea is

that the operators γ†ke, γ†−kh must diagonalize HMF−µN and create elementary quasi-particle excitations ofenergy ω(k) (electrons) and ω(−k) (holes), respectively,if applied to the ground state. Therefore the equationsof motion for the operators γke and γ−kh are

ihγke =[

γke,HMF− µN]

= ω(k) γke,

ihγ−kh =[

γ−kh,HMF− µN]

= ω(−k) γ−kh, (24)

and the Hamiltonian HMF− µN acquires the form

HMF− µN =∑

k

[

ω(k) γ†keγke

+ ω(−k)γ†−khγ−kh

]

. (25)

Note that in our definition, Eqs. (24), the operator γ†−khis a fermionic creation operator which excites a hole,hence the hole energy, as well as the electron energy,is positive, i.e. ω(k) > 0, ω(−k) > 0 [“excitation rep-resentation” (ER)]. Besides, ω(k), ω(−k) depend on thechemical potential µ and on the built-in coherence ∆,and consequently on the temperature T . The γ opera-tors are analogous to the Bogoliubov-Valatin operatorsof the BCS theory; however, in the present context theeigenstates of the hamiltonian HMF have definite num-

bers of particles, namely γ†ke creates an electron and γ†−khcreates a hole, while the particle number of the BCS so-lution is an average quantity, and in that case γ† createsa quasi-particle which is a mixture of electron and hole.

6

From Eqs. (24) and the condition that the coefficients of each independent γ-operator must be zero, we obtainfinite-difference equations for the electron site-coefficients u and v of the junction. For the bulk semiconductor theBdG equations are:

[ω(k) + µ]uk(j) = −tuk(j − 1)− tuk(j + 1)−(

∆∗ − V ∗df)

vk(j − 1/2)−(

∆∗ + V ∗df)

vk(j + 1/2) ∀j > 1,

[ω(k) + µ] vk(j + 1/2) = − (∆ + Vdf )uk(j)− (∆− Vdf ) uk(j + 1) ∀j > 0. (26)

Similarly, the hole amplitudes u and v are given by:

[ω(−k)− µ] u−k(j) = tu−k(j − 1) + tu−k(j + 1) + (∆− Vdf ) v−k(j − 1/2) + (∆+ Vdf ) v−k(j + 1/2) ∀j > 1,

[ω(−k)− µ] v−k(j + 1/2) =(

∆∗ + V ∗df)

u−k(j) +(

∆∗ − V ∗df)

u−k(j + 1) ∀j > 0. (27)

To solve Eq. (26), we write down the trial two-component wavefunction

(

uk(j)

vk(j + 1/2)

)

=

(

uk

vkeika/2

)

eikaj . (28)

In the following, we take without loss of generality uk >0, and ∆, Vdf (∆L, ∆R, VdfL, VdfR) real. Equation (28)is compatible with (26) only if

ω(k) = ξk ± Ek − µ, (29)

where

ξk = εk/2, εk = −2t cos (ka),

Ek =

√

ξ2k + |∆k − Vk|2,

∆k = 2∆cos (ka/2), Vk = 2iVdf sin (ka/2). (30)

Since ω > 0, we choose the sign + in Eq. (29). Conse-quently, the amplitudes (uk, vk) are given by

(Vk −∆k) uk = (ξk + Ek) vk, (31)

plus the normalization condition

u2k + |vk|2 = 1, (32)

that is

u2k =

1

2

(

1 +ξkEk

)

, |vk|2 =1

2

(

1− ξkEk

)

, (33)

and the phase of vk is determined by Eq. (31). In thesame way, the solution for holes is (u−k > 0)

(

u−k(j)

v−k(j + 1/2)

)

=

(

u−kv−ke−ika/2

)

e−ikaj , (34)

with energy

ω(−k) = −ξ−k + E−k + µ, (35)

where

E−k =√

ξ2−k + |∆−k + V−k|2, (36)

and amplitudes

u2−k =

1

2

(

1− ξ−k

E−k

)

,

|v−k|2 =1

2

(

1 +ξ−kE−k

)

, (37)

and the phase of v−k determined by the equation

(

ξ−k + E−k)

u−k = (∆−k + V−k) v−k. (38)

Figure 2 shows typical quasi-particle band structures onboth sides of the junction for the three different cases un-der study. In this picture the energy branches are drawnaccording to the “semiconductor representation” (SCR),where the quasi-particle energy ǫ for holes is negative,namely

ǫ ={

ω for electrons−ω for holes.

In SCR the ground state (vacuum of γ-operators in ER)can be represented regarding the valence band as all filledwith electrons and the conduction band empty. Note thatthe asymmetry of conduction and valence bands close tothe gap is remarkable in case (i) and (ii), contrary to case(iii). For FE (NB) the gap is indirect, and the bottom(top) of conduction (valence) band is much flatter thanthe top (bottom) of valence (conduction) band, while thecurvature of the two SC bands close to the direct gap iscomparable. Besides, as a consequence of the transforma-tion (21), the mapping k → π/a− k, ǫ → −ǫ transformsthe FE electron (hole) band into the NB hole (electron)band.Once we know the semiconductor bulk energy spec-

trum, we can also compute the probability current den-sity JkeN (j) [J−khN (j)] associated with the quasi-particlewavefunction (u, v) [(u, v)] (see appendix A). FromEq. (A11-A14), (28) and (34) we obtain

JkeN (j) =2t

hu2k sin (ka) +

1

huk |vk|

[

(∆ + Vdf )

× sin (ka/2 + ϕ) + (∆− Vdf )

× sin (ka/2− ϕ)]

, vk = |vk| eiϕ, (39)

7

J−khN (j) =2t

hu2−k sin (ka) +

1

hu−k |v−k|

[

(∆ + Vdf )

× sin (ka/2− ϕ) + (∆− Vdf )

× sin (ka/2 + ϕ)]

, v−k = |v−k| eiϕ. (40)

Clearly the bulk current is site-independent. It can beshown that

JkeN =1

ah

∂ ω(k)

∂k, J−khN =

1

ah

∂ ω(−k)

∂ (−k),

(41)i.e. the quasi-particle velocity has the same expressionas the semi-classical one. According to formulae (41) forthe probability density current, the solution (28) [(34)]represents a two-component electron (hole) wavefunctionwith wave vector k (−k) travelling from left to right if k >0. In the bulk we assume periodic boundary conditionsfor k and restrict its values to the first Brillouin zone,namely k = (2π/a)n/Ns, −Ns/2 < n ≤ Ns/2, Ns even.Let us focus on case (i). In order to compute the tem-

perature dependence of the order parameter ∆, recalldefinition (10) and use the inverse transformation of (23)to obtain

∆ =Udf

Ns

∑

k

{

u−k(j) v∗−k(j + 1/2)

⟨

γ−khγ†−kh

⟩

+ u∗k(j) vk(j + 1/2)⟨

γ†keγke

⟩}

. (42)

The average occupations⟨

γ†keγke

⟩

,⟨

γ†−khγ−kh

⟩

are sim-

ply given by the Fermi function f(ω)

⟨

γ†keγke

⟩

= f(ω(k)) ,⟨

γ†−khγ−kh

⟩

= f(ω(−k)) ,

(43)with

f(ω) =(

eβω + 1)−1

β = 1/kBT.

Replacing the (u, v) amplitudes with the values we havejust computed, Eq. (43) turns into

∆(T ) =Udf

Ns

∑

k

{ ∆k

2Ekcos (ka/2)

×[

1− f(ω(−k))− f(ω(k))]}

, (44)

which is the analogous of the BCS gap equation, im-plicitely defining ∆(T ). The restraint (20) on the elec-tron number is an implicit definition of µ(T ); in terms ofγ operators it turns into:

1

Ns

∑

k

f(ω(k)) =1

Ns

∑

k

f(ω(−k)) . (45)

We solve both Eqs. (44) and (45) simultaneously to ob-tain the value of ∆(T ) and µ(T ) deep inside the FEbulk. A critical temperature TC exists beyond which∆ = µ = 0 and the energy gap of the material vanishes.

Since we know the bulk solution on the semiconductorside of the junction, we now study the motion of quasi-particles along the whole junction. The BdG equationsfor the bulk of the metal on the left side

[ω(k) + µ]uk(j) = eV uk(j)

−t′uk(j − 1)− t′uk(j + 1) ,

[ω(k) + µ] vk(j + 1/2) = 0 ∀j < 0, (46)

[ω(−k)− µ] u−k(j) = −eV u−k(j)

+t′u−k(j − 1) + t′u−k(j + 1) ,

[ω(−k)− µ] v−k(j + 1/2) = 0 ∀j < 0, (47)

have the Bloch solution(

uk(j)

vk(j + 1/2)

)

=

(

1

0

)

eiqaj ,

(

u−k(j)

v−k(j + 1/2)

)

=

(

1

0

)

e−iqaj , (48)

with energy

ω′(q) = eV − 2t′ cos (qa)− µ,

ω′(−q) = −eV + 2t′ cos (−qa) + µ, (49)

and, from Eq. (A9-A10), probability current density

JkeN (j) =2t′

hsin (qa),

J−khN (j) = −2t′

hsin (−qa), (50)

or, equivalently,

JkeN =1

ah

∂ ω′(q)

∂q, J−khN =

1

ah

∂ ω′(−q)

∂ (−q). (51)

The idea is to match in some way the bulk solutions onboth sides of the junction. To make the discussion easier,let us consider only the electron motion. The physicalboundary condition is that the electron travels e.g. fromz = −∞ towards positive values of z and it is partlytransmitted through the junction and partly reflected.Note that the same energy ω corresponds to two differentbulk wave vectors q and k, while in metal/superconductorjunctions the wave vector can be approximated by theFermi vector on both sides of the interface. We alwaysuse k to label the coherent electronic state through thewhole space, with the convention that both k and q, wavevectors in their respective bulks, correspond to the sameenergy ω. It is convenient to define the wavefunctions

(

Ψ1L(j)

Ψ2L(j + 1/2)

)

=

(

1

0

)

eiqaj −Rk

(

1

0

)

e−iqaj ∀j, (52)

(

Ψ1R(j)

Ψ2R(j + 1/2)

)

= Tk

(

uk

vkeika/2

)

eikaj ∀j, (53)

8

with the elastic scattering condition

ω′(q) = ω(k) . (54)

If we define(

uk(j)

vk(j + 1/2)

)

as the solution of the motion across the whole junction∀j, we immediately have from Eq. (46) that

(

uk(j)

vk(j + 1/2)

)

=

(

Ψ1L(j)

Ψ2L(j + 1/2)

)

∀j < 0,

uk(0) = Ψ1L (0) , (55)

and from Eq. (26) that

(

uk(j)

vk(j + 1/2)

)

=

(

Ψ1R(j)

Ψ2R(j + 1/2)

)

∀j > 0, (56)

because (26) and (46) are linear and homogeneous. Wehave still to determine vk(1/2) and the two unknown con-stants Tk and Rk, so we need the three BdG equationsat the interface not yet employed:

[ω(k) + µ]uk(0) = εd0uk(0)− t′uk(−1)− tuk(1)−(

∆∗L + V ∗dfL

)

vk(1/2) , (57)

[ω(k) + µ] vk(1/2) = εf1/2vk(1/2)−(

∆L + VdfL

)

uk(0)−(

∆R − VdfR

)

uk(1) , (58)

[ω(k) + µ]uk(1) = −tuk(0)− tuk(2)−(

∆∗R − V ∗dfR

)

vk(1/2)−(

∆∗ + V ∗df)

vk(3/2) . (59)

We replace uk(−1), uk(0), uk(1), vk(3/2), uk(2) in (57-59) with Ψ1L(−1), Ψ1L(0), Ψ1R(1), Ψ2R(3/2), Ψ1R(2), respec-tively, obtaining a linear system for the three unknown quantities vk(1/2), Rk, and Tk:

−(

∆L + VdfL

)

Rk +(

ω − εf1/2)

vk(1/2) +(

∆R − VdfR

)

ukeikaTk = −

(

∆L + VdfL

)

−tRk +(

∆∗R − V ∗dfR

)

vk(1/2) +(

ωukeika + tuke

2ika +(

∆∗ + V ∗df

)

vke(3/2)ika

)

Tk = −t(

−ω + εd0 − t′eiqa)

Rk +(

∆∗L + V ∗dfL

)

vk(1/2) + tukeikaTk = −ω + εd0 − t′e−iqa.

(60)

To solve system (60) we fix ω, then we invert Eqs. (29) and (49) to obtain k and q, and hence uk and vk: once we

input as parameters ∆L, ∆R, VdfL, VdfR, εd0, and εf1/2, the whole coefficient matrix of system (60) is known and Tk

and Rk can be eventually obtained.

In Eq. (52) we chose a normalization such that the fluxof the incident wave J inc

keN (j) is

J inckeN (j) =

2t′

hsin (qa) =

1

ah

∂ ω′(q)

∂q∀j < 0, (61)

and that of the reflected wave J reflkeN (j) is

J reflkeN (j) = − |Rk|2

2t′

hsin (qa) ∀j < 0, (62)

hence, by definition, the reflection coefficient R(ω′(q)) =R(ω(k)) is given by

R(ω) =

∣

∣J reflkeN

∣

∣

∣

∣J inckeN

∣

∣

= |Rk|2 . (63)

The transmission coefficient T (ω) can be calculated bythe definition

T (ω) =|J trans

keN |∣

∣J inckeN

∣

∣

, (64)

with

J transkeN (j) =

2t

h|Tk|2 u2

k sin (ka) +1

h|Tk|2 uk |vk|

×[

(∆ + Vdf ) sin (ka/2 + ϕ)

+ (∆− Vdf ) sin (ka/2− ϕ)]

∀j > 0, (65)

or, more simply, by probability conservation

T (ω) = 1−R(ω) . (66)

T (ω) is the key quantity we need to compute the ther-mopower.Because of the two-fold degeneracy of energy ω, we

have to consider also the quasi-particle associated with−k (and −q), with k > 0. It is clear how to rewriteEq. (52-53), because now the electron, coming from theright, is partly reflected into the right-hand side of the

9

junction and partly transmitted into the left-hand side,i.e.

(

Ψ1L(j)

Ψ2L(j + 1/2)

)

= T−k

(

1

0

)

e−iqaj , (67)

(

Ψ1R(j)

Ψ2R(j + 1/2)

)

=

(

u−kv−ke−ika/2

)

e−ikaj

− R−k

(

u−kv−keika/2

)

eikaj . (68)

After that, we can proceed in the same way as above. Bytime-reversal symmetry, one has

T (ω(k)) = T (ω(−k)) . (69)

An analogous ER procedure is used to compute the trans-mission and reflection coefficients T (ω) and R(ω) forholes. In this last case, Eqs. (52-53) turn into (withq, k > 0)

(

Ψ1L(j)

Ψ2L(j + 1/2)

)

=

(

1

0

)

e−iqaj − R−k

(

1

0

)

eiqaj , (70)

(

Ψ1R(j)

Ψ2R(j + 1/2)

)

= T−k

(

u−kv−ke−ika/2

)

e−ikaj . (71)

The above derivation of the transmission coefficientstill holds for case (iii) as long as we rename and setthe parameters as discussed in section II. Some cautionis needed in taking the correct sign of the wavevector kfor certain values of parameters outside the range we ac-tually examined. Namely, if 2t2(∆2 +V 2

df ) < (∆2 −V 2df )

2

and t2 < ∆2 −V 2df the electron wavefunction (28) travels

from right to left if k > 0, like the hole wavefunction (34)when 2t2(∆2 + V 2

df ) < (∆2 − V 2df )

2 and t2 < V 2df −∆2.

B. Transport coefficients and ZT

If we apply an electric field E or a temperature gradient∇T across the junction, we produce electric and thermalcurrents JE and JT , respectively. In a stationary statethe relation between fluxes and driving forces is linear, iffields are small enough:37

{

JE = LEEE + LET∇TJT = LTEE + LTT∇T.

(72)

The transport coefficients L are not all independent, be-cause of the Onsager relation

LET = −LTE

T. (73)

The coefficients are related to the electrical and thermalconductivities σ and κ,38 respectively, and to the ther-mopower (Seebeck coefficient) Q of the junction:

σ = LEE, κ = −(

LTT − LTELET

LEE

)

,

Q = −LET

LEE. (74)

A high value of Q alone does not necessarily imply thatthe junction is a good thermoelectric cooler, that is whyengineers introduce the figure of merit Z:

Z =Q2σ

κ. (75)

Indeed, if we pump heat from the cooler side to the hot-ter one, we need high pumping efficiency (high Q), lowproduction of heat through Joule heating (high σ), andlow backwards conduction of heat (low κ).6 Z has unitsof inverse temperature, thus in general what is quoted isthe dimensionless quantity ZT . We shall also use anothernatural definition of figure of merit, ZT/ (1 + ZT ), whichis the dimensionless figure, defined below in Eq. (93),more directly related to the transmission coefficient mo-menta we compute.Since the computation of the electrostatic and thermal

field across the junction is complicated and not essen-tial to the junction thermopower, we simply assume thatthe voltage V and the temperature T are constant onboth sides of the junction and have a sharp step at theinterface. Thus, instead of Eq. (72), we write

{

JE = KEE (−δV ) +KET δTJT = KTE (−δV ) +KTT δT,

(76)

where we have replaced the gradients E = −∇V and ∇Twith (−δV ) and δT , respectively, where δV = V (right)−V (left) and δT = T − Tn, being T the temperature as-sociated to the right-hand side, and Tn to the normalmetal on the left-hand side. Since we have assumedV (right) = 0, we have −δV = V (left) = V . The co-efficient KEE represents now the conductance G in placeof the conductivity σ and − (KTT −KTEKET/KEE) thethermal conductance GT

39 in place of the thermal con-ductivity κ, but all the other relations (73-75) still hold:

KET = −KTE

T, (77)

G = KEE , GT = −(

KTT − KTEKET

KEE

)

,

Q = −KET

KEE, (78)

Z =Q2G

GT. (79)

Note that the Onsager relation (77) is still valid despiteour simplifing assumptions about the V - and T -gradients.From Eq. (76) the formulae for the K transport coeffi-cients follow:

KEE =

(

∂JE∂ (−δV )

)

δT,δV =0

,

KET =

(

∂JE∂ (δT )

)

δT,δV =0

,

10

KTE =

(

∂JT∂ (−δV )

)

δT,δV =0

,

KTT =

(

∂JT∂ (δT )

)

δT,δV =0

. (80)

Here we closely follow the approach of Blonder et

al.40 We assume that the two sides of the junction are

in contact with perfect electron reservoirs, at differenttemperatures and chemical potentials, and that quasi-particle wavefunctions keep their phase coherence acrossthe whole system except at z = ±∞, where they com-pletely loose their phase, thermalize and relax in energydue to inelastic scattering processes of the Fermi sea.Namely, we regard the contacts as perfect emitters oradsorbers. The electric current density, JE , will be givenby the sum of all contributions of the quasi-particle exci-tations to the current, each one weighted by the correctFermi distribution function, depending if quasi-particlesoriginate from the left or right reservoir. Because JE isstationary and conserved, we can calculate it in everypoint of the space, so we choose a lattice site j < 0 in thebulk on the left-hand side. In particular, the contribu-tion to the density current JL→R

E produced by electronsgoing from left to right is [see Eq. (A8)]

JL→RE = 2

1

Ns

∑

k>0

J incke (j) f(ω(k)− eV ) j < 0, (81)

where the Fermi function refers to the temperature Tn ofthe metal on the left-hand side, and the electrochemicalpotential differs from that on the right-hand side by eV .The prefactor 2 accounts for the spin degeneracy. Notethat Eq. (81) gives the only electron flux running fromleft to right on the bulk metal side. On the other hand,the contribution to the electron current JL←R

E from rightto left, always on the bulk metal side, is given by twodistinct terms, one for electrons reflected at the interfaceand coming from the left side, hence in equilibrium withthe left reservoir (with temperature Tn and Fermi func-tion referred to eV ), and one for electrons transmittedthrough the interface and coming from the right reser-voir at temperature T and at ground potential:

JL←RE = 2

1

Ns

∑

k>0

J reflke (j) f(ω(k)− eV )

+ 21

Ns

∑

k<0

J transke (j) f(ω(k)) j < 0. (82)

Note that the second sum in Eq. (82) runs over negative kvalues and the related electron wavefunctions extend allover the junction, being therefore well defined at j < 0.Now we add (81) and (82), using (63) and (64),

JL→RE + JL←R

E = 21

Ns

∑

k>0

J incke (j < 0)

× [1−R(ω(k))] f(ω(k)− eV )

+ 21

Ns

∑

k<0

J incke (j > 0) T (ω(k)) f(ω(k)) . (83)

Here the notation J incke (j > 0) refers to electrons, incident

on the interface, coming from z = +∞. From Eqs. (41)and (51)

JL→RE + JL←R

E = 21

Ns

∑

q>0

e

ah

∂ ω′(q)

∂q

× [1−R(ω′(q))] f(ω′(q)− eV )

+ 21

Ns

∑

k<0

e

ah

∂ ω(k)

∂kT (ω(k)) f(ω(k)) , (84)

and going to the continuum limit∑

k →(Nsa) /(2π)

∫

d k we obtain

JL→RE + JL←R

E =2e

2πh

∫ π/a

0

d q

× ∂ ω′(q)

∂qT (ω′(q)) f(ω′(q)− eV )

+2e

2πh

∫ 0

−π/a

d k∂ ω(k)

∂kT (ω(k)) f(ω(k)) , (85)

or

JL→RE + JL←R

E =2e

h

∫ ∞

0

dω

× T (ω) [f(ω − eV )− f(ω)] , (86)

with the convention that T (ω) = 0 if ω is not in the rangeof excitation energies allowed, and h = 2πh. Similarly,the hole contribution to JE is given by

− 2e

h

∫ ∞

0

d ωT (ω) [f(ω + eV )− f(ω)] . (87)

Now we add (86) and (87), passing to SCR and using theequivalence f(−ǫ) = 1− f(ǫ), to obtain the current JE :

JE =2e

h

∫ ∞

−∞

d ǫ[

T (ǫ) + T (−ǫ)]

× [f(ǫ− eV )− f(ǫ)] . (88)

The electron (hole) heat current density JkeT (j)[J−khT (j)] is defined by:

JkeT (j) = ω(k)JkeN (j) ,

J−khT (j) = ω(−k)J−khN (j) (89)

(note that in ER energies ω are referred to µ). Reason-ing in the same way as for JE , we obtain the total heatcurrent JT :

JT =2

h

∫ ∞

−∞

d ǫ[

T (ǫ) + T (−ǫ)]

× ǫ [f(ǫ− eV )− f(ǫ)] . (90)

Now it is easy to compute K-transport coefficients fromdefinitions (80). We only summarize the results:

KEE =2e2

hL0, KET = − 1

T

2e

hL1,

KTE =2e

hL1, KTT = − 1

T

2

hL2, (91)

11

with

Ln =

∫ ∞

−∞

d ǫ[

T (ǫ) + T (−ǫ)]

ǫn(

−∂ f(ǫ)

∂ ǫ

)

. (92)

From Eq. (91) one can immediately check that the On-sager relation (77) is fulfilled. Besides, note the strikingsimilarity of Eq. (91-92) with the formalism of the semi-classical theory of transport (Boltzmann Equation):37 inour case the role of the k-dependent relaxation time τ(k)is played by the transmission coefficient T , the interfacebeing the scattering mechanism. From the transmissioncoefficient momenta (92) we compute the alternative fig-ure of merit

L21

L0L2=

ZT

ZT + 1. (93)

IV. RESULTS AND DISCUSSION

We present the results for the thermopower Q andthe figure of merit ZT for the different types of junc-tion under study. Bulk parameters are chosen to de-scribe a large class of materials and are the same as inFig. 2, with the metal band much broader than the semi-conductor one. In particular, for FE and NB on theright-hand side of the junction the indirect gap Egap atT = 0 is 0.40t, i.e. one tenth of the “bare” d-bandwidth

4t [Egap = −t +√

t2 + 4∆2(0) and −t +√

t2 + 4V 2df ,

respectively]. In case (iii), the gap is direct, Egap =0.40t, where 4t is roughly the total bandwidth (electron

plus hole band) of SC [Egap = 2√

t′′2 + 4V 2e ]. With this

choice of parameters, the three semiconductors have thesame gap at T = 0 and approximately the same band-width.

A. Clean interface

First we study the junction with a clean interface.With this terminology we mean that there are no d- or f -impurity layers (εd0 = εf1/2 = 0) at z = 0 or z = a/2, re-spectively. For case (i) we assume that the change in theorder parameter ∆(T ) at z = 0 is abrupt, from zero to the

bulk value [∆L = ∆R = ∆(T )]. Similarly, we take the hy-bridization or hopping parameters at the interface equalto the bulk values in both cases (ii) [VdfL = VdfR = Vdf ]

and (iii) [VeL = VeR = Ve, tL = tR = t].In Fig. 3(a) we plot the absolute value of thermopower

|Q| vs temperature. In all three cases |Q| goes to in-finity as T → 0, as expected for both indirect-gapnarrow-band semiconductors41 and direct-gap ordinarysemiconductors.37 As T approaches the critical tempera-ture Tc of the ferroelectric (kBTc =0.195t for our choiceof parameters), the thermopower |Q| of FE goes to zero,while |Q| for NB and SC decreases in a exponential-likemanner as the temperature rises. At T = TC the FE gap

0.00 0.05 0.10 0.15KBT (t units)

10−6

10−5

10−4

10−3

10−2

10−1

ther

mop

ower

|Q| (

V/K

)

FENBSC

0.05 0.10 0.15 0.20 0.25 0.30 KBT (t units)

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

ther

mop

ower

Q (

mV

/K) FE

NBSC

(i)

(iii)

(ii)

0.00 0.05 0.10 0.15 KBT (t units)

0

2

4

6

figur

e of

mer

it Z

T (

dim

ensi

onle

ss)

FENBSC

(i)

(iii) (ii)

0.00 0.05 0.10 0.15 KBT (t units)

0.0

0.2

0.4

0.6

0.8

1.0

ZT

/(Z

T+

1) (

dim

ensi

onle

ss)

FENBSC

(i)

(iii)(ii)

(a)

(b)

FIG. 3: (a) Absolute value of the thermopower |Q| vs T , forthe three cases under study. |Q| is in logarithmic scale. Notethat |Q| of FE drops to zero, as T → Tc (Tc = 0.195t, withparameters of Fig. 2). Inset: Magnification of the same plot,in linear scale, in the neighborhood of Tc for FE. If T > Tc

then Q = 0 for FE, because above the critical temperaturethe gap vanishes and the bulk FE turns into a metal [T (ǫ) =T (−ǫ)]. (b) Plot of the corresponding figure of merit ZT vs T .Like Q in Fig. 3(a), ZT of FE goes to zero as T → Tc. Inset:same plot for the alternative figure of merit ZT/ (ZT + 1),whose upper bound (ideal thermoelectric) is 1.

vanishes and the semiconductor turns into a metal withsymmetric bands with respect to µ, i.e. Q = 0. In thelow temperature region, instead, for kBT ≤ 0.05t (T ≤0.25Tc), i.e., in the region where ∆(T ) is almost constant,far from the second-order ferroelectric/metal transition,the absolute value of Q for FE and NB is nearly iden-tical. For SC |Q| is about 60%-70% of the NB value inthe whole range of temperature. To compare our resultswith reported bulk data, we can assume a typical value

12

0.0 0.4 0.8

−2

−1

0

1

2

ener

gy ε

(t u

nits

)

0.0 0.4 0.8

05 t

0.0 0.4 0.8

10 t20 t

(i) (ii) (iii)

d−impurity

T

(

�

)

+

�

T

(

��

)

FIG. 4: Total (electron plus hole) transmission coefficientT (ǫ) + T (−ǫ) in “semiconductor representation” (SCR) as afunction of the energy ǫ (vertical axis). From left to rightpanels correspond to FE, NB, and SC, respectively. Curvesfor different values of the d-impurity energy εd0 are plotted.Note that T (ǫ) of FE is mapped into T (−ǫ) of NB underthe transformation ǫ → −ǫ, as a consequence of equations(21). The d-impurity energy for four curves is indicated inthe diagram above. Other parameters used are the same asin Fig. 2 and T = 0.

of 4t = 1 eV for FE or NB, i.e. a gap around 0.1 eV atT = 0, with room temperature ∼ 0.1t. With these num-bers, we have |Q| = 0.11 – 0.14 mV/K at room temper-ature, values comparable with those of some rare-earthmetals with very high thermopower. If instead we as-sume for the broad band semiconductor SC a typical gapof 0.5 eV, then |Q| is around 0.5 mV/K at room tem-perature, consistently with characteristic bulk data. Theinset of Fig. 3(a) is a magnification of the plot (in lin-ear scale) in the neighborhood of Tc. The discontinuityof the derivative of Q for the FE curve is a signature ofthe second-order ferroelectric transition: if T > Tc thenQ = 0. Similar kinks are found also for the other trans-port coefficients G and GT . One sees that NB and SChave electron transport character (Q < 0) while FE hashole character (Q > 0).To understand the above features, in Fig. 4 we plot

the total (electron plus hole) transmission coefficientT (ǫ)+ T (−ǫ) vs energy in SCR for the three cases understudy (at T = 0). Solid lines refer to the clean interfacecase. Since the sign and magnitude of Q is establishedby Eq. (92) for L1, i.e. by the competition between thedifferent weights of electron and hole transmission func-tions T and T , respectively, it is clear that the strongerthe electron/hole asymmetry of the transmission coeffi-cient, the higher the thermopower. Panels (i) and (ii) ofFig. 4 present the remarkable electron/hole asymmetryclose to the gap, hence Q for FE has a strong hole charac-ter (Q > 0) while NB has a dominant electron character(Q < 0). The mapping of T (ǫ) of FE into T (−ǫ) of

NB under the transformation ǫ → −ǫ —a consequence ofthe transformation (21)— explains why |Q| is the samefor FE and NB at low T . The situation is different forSC [see panel (iii)], because electron and hole transmis-sion coefficients are more symmetric, and thus |Q| hasa lower value. In the limit of electron/hole symmetry[T (ǫ) = T (−ǫ), as it is the case for FE when T → Tc],the thermopower is zero.The other transport parameters, apart from Q, are the

electrical conductance G and the thermal conductanceGT . We find that, for all the three cases under study,there exists an activation temperature around 0.03t, be-low which G and GT , as functions of T , rapidly (ex-ponentially) drop to zero, because the number of ther-mally excited carriers becomes too small. This behav-ior, characteristic of a material with gap, shows that atlow temperature the main contribution to the thermalconductance GT is given by the lattice, which is not in-cluded, thus dramatically decreasing the actual value ofZT . This contribution, and hence the minimum work-ing temperature of the junction, depends on the thermalimpedance mismatch of the interface.Once all transport coefficients are known, the most

relevant quantitity to be computed for practical appli-cations is the figure of merit ZT , plotted in Fig. 3(b) asa function of the temperature. In all three cases ZT isa monotonic decreasing function of T . However, ZT ismuch bigger for FE and NB than for SC (but if T → Tc

then ZT → 0 for FE). With the numerical parameters wehave employed above, at room temperature ZT is ∼ 0.3for FE and ∼ 0.4 for NB, but already at T =100 K ZT is∼ 1.1 for FE and NB and only ∼ 0.5 for SC. In the insetof Fig. 3(b), we redraw the same plot in term of the alter-native figure of merit ZT/ (ZT + 1): this is the quantitywe will examine in the following. While ZT has no the-oretical upper bound, the maximum of ZT/(ZT + 1) is1, corresponding to ZT = ∞.It is likely that the one-dimensional model artificially

enhances ZT . Besides, we do not know how the orderparameter ∆(T ) actually varies at the interface, and allthe effects of charge polarization are neglected: this doesnot necessarily imply a reduction of ZT . The only scat-tering mechanism we have considered is the interface: inparticular, the contribution of phonons to the thermalconductance is neglected.

B. d-impurity layer

Now we study the effect of an overlayer of d-impurityatoms at the interface. This structure could be built byepitaxial growth techniques: we find that this configu-ration does not improve considerably the figure of meritZT .In our one dimensional virtual crystal model, we de-

scribe for simplicity the overlayer by putting one atomat z = 0 with site energy εd0. Since in the calculationwe leave unchanged the hopping coefficient t and all the

13

other parameters with respect to the clean-interface case,we also require, from a physical point of view, that theatomic orbital of the impurity is still of d-type. The es-sential point here is that the atom at z = 0 substantiallyparticipates in the electronic motion, contrary to a f -site.In case (iii) this distinction between f - and d-sites is nolonger relevant: actually we look at SC only for compar-ison. One could also think of growing an overlayer atz = 0 only partially filled with impurities: in that casethe values of transport coefficients should be an inter-polation between the two limiting values correspondingto the cases of 0% (clean surface) and 100% impurityconcentration within the layer.To understand the role of the d-impurity layer in the

transport, consider the transmission in Fig. 4. Here, aswell as the total transmission coefficient vs energy for theclean interface (solid lines), we have also plotted curvescorresponding to increasing values of the impurity levelεd0. As the impurity level energy εd0 increases, the trans-mission is uniformly depressed over the whole range ofenergies. In case (i) and (ii), results depend only on theabsolute value of εd0, while (iii) is more complex, due tothe presence of a second nearest neighbor hopping coef-ficient.While the depression effect of the impurity seems im-

portant for both NB and FE semiconductors, some cau-tion is needed in accepting these results. First, one can-not make |εd0| arbitrarily large, because its value is physi-cally limited and cannot differ too much from typical bulkenergies, otherwise the impurity would be screened. Sec-ond, in our computation we have taken all parameters butεd0 unchanged with respect to the clean-interface case,and it is clear that this approximation becomes worse aslong as |εd0| increases. For example, ∆l or VdfL wouldsurely change as εd0 varies.The conclusion is that the enhancement of the figure of

merit ZT/ (ZT + 1) for reasonable values of εd0 is quitelimited. In general, as long as we increase the magnitudeof εd0, we slightly enhance ZT uniformly over the wholerange of temperatures. It is remarkable that FE and NBvalues of ZT are always much higher than SC values. Forexample, at room temperature, with the usual choice ofnumerical parameters and εd0 = 5t, ZT ∼ 0.4 for FE,∼ 0.6 for NB, and ∼ 0.2 for SC, but already at 100 KZT ∼ 1.8 for FE and NB, while it is only ∼ 0.4 for SC.

C. f-impurity layer

In order to dramatically improve the figure of merit, wepropose the insertion of a rare-earth overlayer at z = a/2,with f -level energy εf1/2. Contrary to the d-impuritylayer discussed in the previous section, here it is essentialthat the localized impurity level is of f -type, i.e. hardlysharing the electronic conduction. This is not the casefor SC, where tunneling from this “f -site” to adjacentneighbors is allowed: in fact, for SC, the situation is ba-sically similar to the previous d-impurity case. In case

0.0 0.4 0.8

−2

−1

0

1

2

ener

gy ε

(t u

nits

)

0−0.4t

0.0 0.4 0.8

−0.7t−1.5t

0.0 0.4 0.8

(i)

(ii) (iii)f−impurity

T

(

�

)

+

�

T

(

��

)

FIG. 5: Total (electron plus hole) transmission coefficientT (ǫ) + T (−ǫ) in SCR as a function of the energy ǫ (verticalaxis). From left to right vertical panels correspond to FE,NB, and SC, respectively. Curves for different values of the f -impurity energy εf1/2 are plotted. A consequence of equations(21) is that T (ǫ) of FE for εf1/2 > 0 can be mapped intoT (−ǫ) of NB under the transformation −ǫ → ǫ, −εf1/2 →εf1/2. The parameters used are the same as in Fig. 2 withT = 0.

(iii), hopping to adjacent neighbors is allowed both fromthe “f -site” and from the “d”-site. In fact, SC results forthe figure of merit ZT are not dissimilar from the valuesobtained in the d-impurity case. In order to gain someinsight into the thermopower behavior, in Fig. 5 we haveplotted the total transmission coefficient T (ǫ) + T (−ǫ)vs ǫ for different values of εf1/2, similar to Fig. 4. FEand NB curves [panel (i) and (ii), respectively] are quali-tatively different from those of SC [panel (iii)], as we setεf1/2 to negative values in the energy band range. For FE

and NB, T goes to zero in the neighborhood of εf1/2, asif the hole were completely backscattered from the inter-face when resonates the energy with that of the impurityatom. On the contrary, in case (iii) the effect is opposite,with T gaining weight for energies close to εf1/2. Thetrend is similar for εf1/2 > 0, showing total reflectionin the neighborhood of positive values of εf1/2, in case(i) and (ii). These results demonstrate that while thef -impurity level of FE and NB does not share the elec-tronic conduction, due to localization, it strongly affectsthe transmission, because either in case (i) the quasi-particle excitation is a coherent superposition of d- andf -states, or in case (ii) the “bare” bands are hybridized.The case of SC is excluded here.

The overall effect of the f -impurity layer on transportis so strong that it even changes the dominant (electronor hole) character of the thermopower, i.e. the sign of Q.For the sake of simplicity, we now consider only FE. InFig. 6 we plot Q vs T for different values of εf1/2. Leftpanel refers to εf1/2 < 0. We see that, as we set εf1/2from zero to negative values, Q changes sign: already atεf1/2 = -0.1t Q has electron character (Q < 0) as T → 0,

14

0.00 0.05 0.10 0.15KBT (t units)

−0.3

−0.2

−0.1

0

0.1

0.2

0.3th

erm

opow

er Q

(m

V/K

)

0−0.075 t−0.1 t−0.125 t−0.2 t

0.05 0.10 0.15KBT (t units)

00.175 t0.25 t0.5 t0.75 t0.9 t

f−impurityf−impurity

(i) FE (i) FEεf1/2<0 εf1/2>0

FIG. 6: Thermopower Q vs T for FE. Left panel: Curvesfor different values of the f -impurity energy εf1/2 (negative).Right panel: The same, with εf1/2 > 0. Note that Q changessign for some curves as T varies, and that |Q| → ∞ as T → 0.Bulk parameters as in Fig. 2.

while for kBT > 0.02t Q has hole character (Q > 0).This behavior can be understood by examining Fig. 5:the impurity level drastically diminishes the weight of thetransmission coefficient at the top of the valence band,while increasing it at the bottom of the conduction band,so that the sign of Q is reversed at low temperatures,where the only excited carriers are those whose energiesare close to the gap. Note that Q is extremely sensitiveto the position of εf1/2, in contrast to the case of thed-impurity layer. The right panel of Fig. 6 presents asimilar situation for εf1/2 > 0. Here the situation is alittle less obvious: as we raise the value of εf1/2 first Qrises then drops and changes sign. This is due to theasymmetry of electron and hole bands. At first εf1/2,now positive, is close to the bottom of the conductionband, favoring the hole transport because it depressesthe thermally activated electronic channels. Then, asεf1/2 is increased, weight is added to the transmissioncoefficient at the bottom of the conduction band. Thisweight is “swept away” from the energy neighborhoodresonant with εf1/2, that now is higher in energy withrespect to the band bottom. This mechanism also makesQ change sign.

To study the behavior of ZT as the f -impurityεf1/2 varies, we take a series of “snapshots” depictingZT/ (ZT + 1) vs T for different values of εf1/2 in Fig. 7.Figure 7(a) focuses on εf1/2 < 0, in a range between0 and -0.2t. We see that, as εf1/2 is lowered from itszero value, ZT/ (ZT + 1) conspicuously decreases: inthis range Q changes sign. At εf1/2 = -0.1 and -0.125ta temperature exists at which ZT = 0: these tem-peratures correspond to the zero of Q in Fig. 6 (leftpanel). At εf1/2 = -0.2t, however, ZT/ (ZT + 1) be-gins to rise. In Fig. 7(b) we analyze the “high-valueregion” of ZT/ (ZT + 1). At εf1/2 = -0.24t the curve

0.00 0.05 0.10 0.15 KBT (t units)

0.0

0.2

0.4

0.6

0.8

1.0

ZT

/(Z

T+

1) (

dim

ensi

onle

ss)

0−0.05 t−0.1 t−0.125 t−0.2 t

f−impurity

(i) FE

0.00 0.05 0.10 0.15 KBT (t units)

0.0

0.2

0.4

0.6

0.8

1.0

ZT

/(Z

T+

1) (

dim

ensi

onle

ss)

−0.24 t−0.32 t−0.4 t−0.5 t−0.75 t

f−impurity

(i) FE

(a)

(b)

FIG. 7: (a) ZT/ (ZT + 1) vs temperature T for differentvalues of the f -impurity level energy εf1/2 for FE. Bulk pa-rameters as in Fig. 2. (b) The same as Fig. 7(a), for lowervalues of εf1/2. Note that the “shoulder” in the curves isshifted towards higher values of T , as εf1/2 is decreased. Thedashed line represents the envelope of the curves.

is quite depressed around kBT = 0.1 (room temperaturewith the usual numerical parameters), but it acquires gi-ant values around T = 0, close to the theoretical upperbound 1. Moreover, the shape of the curve is very differ-ent from the typical pattern of the d-impurity layer: inthis latter case the curve is convex around T = 0, whilein the present situation ZT/ (ZT + 1) is concave, i.e. ifone slightly departs from T = 0 ZT will still be veryhigh. This approximate flatness of the curve suggeststhat a stable low-temperature working point should ex-ist for a junction-based device. As εf1/2 is furtherly de-creased, ZT/ (ZT + 1) shows a concave “shoulder” whichis shifted towards higher temperatures. This feature im-plies that, once we fix a temperature, it is possible to findan optimal value of εf1/2 maximizing the figure of merit.

15

0.00 0.05 0.10 0.15 KBT (t units)

0.0

0.2

0.4

0.6

0.8

1.0op

timal

ZT

/(Z

T+

1) (

dim

ensi

onle

ss)

εf1/2<0εf1/2>0

f-impurity sign

FE

(ii)

(i)NB

FIG. 8: Maximum value of ZT/ (ZT + 1) vs T for an opti-mized choice of the f -impurity level energy εf1/2. Case (i)(solid lines) and (ii) (dot-dashed lines) are represented. Theup (down) triangles refer to the positive (negative) values oflocal maxima εf1/2, computed at fixed T . Note that the solidline with down triangles is the envelope curve of Fig. 7(b).Bulk parameters as in Fig. 2.

This locus of optimal working points is manifestly givenby the envelope curve for the “shoulders” (dashed line).The behavior of ZT/ (ZT + 1) as εf1/2 rises from zero(εf1/2 > 0) is similar, but the sequence is inverted withrespect to the case we have just examined: this time firstZT/ (ZT + 1) dramatically increases, and then rapidlydrops to low values.

Figure 8 shows the maximum values of ZT/ (ZT + 1)vs T that can be reached with a suitable f -impurity en-ergy εf1/2. Up (down) triangles indicate positive (neg-ative) optimal energies εf1/2. Analogous results for NB(dot-dashed lines) are shown. The curves correspond tothe loci of the T -εf1/2 space like the envelope curve ofFig. 7(b). From Fig. 8 we can derive the absolute max-ima of ZT . Our results are that, at room temperature(kBT = 0.1t, with the usual numerical parameters) ZT ∼1 for FE and 1.6 for NB, at T = 150 K ZT is ∼ 6 in bothcases, at T = 100 K is ∼ 13, and at T = 40 K is ∼ 100.

These extremly high values suggest the possibility ofengineering a periodic lattice of δ-layers made of metaland strongly correlated semiconductor to create an effi-cient thermoelectric device. Borrowing an intuition ofRefs. 14 and 15, the bias applied to the superlatticeshould be such that the electronic transport occurs per-pendicular to interfaces. The thickness of each layershould be comparable to the electronic mean free path,and large enough to prohibit electrons from tunnelling.In this regime the electronic motion within each layeris ballistic, and the interfaces can be considered as theonly scattering mechanisms. Mahan and Woods14 callthermionic a similar device with an ordinary semicon-

ductor replacing the strongly correlated one, because theelectronic motion is ballistic and the expression for thecurrent is the Richardson’s equation. However, here thesituation is different, because the transmission coefficienthas such a sharp variation with energy that the currenthas a more complex form than the Richardson’s expres-sion and it requires a full analysis of the role of the in-terface in transport. The idea here is to grow a f -atomoverlayer at z = a/2 with optimal f -level energy εf1/2depending on the working temperature of the device. Inaddition, we find that the magnitude of thermal conduc-tance changes only slightly with εf1/2. The δ-layer ofrare-earth atoms could also act like an additional sourceof strong scattering for phonons, thus lowering the lat-tice thermal conductance.1 Therefore the junction is apromising candidate as a thermoelectric device.

D. Effects of the surface on the lattice structure

We briefly comment on the influence which the inter-face has on the ideal lattice structure. We consider thenarrow band semiconductor NB. If the effect of the sur-face at z = 0 is to locally shorten the lattice constanta, this will certainly change the hybridization parame-ter VdfL between the d-site at z = 0 and the f -site at

z = a/2. Presumably VdfL will be increased. Modifying

nothing but VdfL with respect to the clean interface case,we find that ZT is only slightly affected by this relaxationeffect. In particular, to change VdfL up to 10% almostrigidly shifts ZT/(ZT + 1) over the whole temperaturerange by approximately 10 – 15 %. From a check of thetransmission coefficient, we find that this surface effectrepresents only a minor perturbation to the electronictransport across the junction.

V. CONCLUSIONS

We have made a qualitative theoretical study of thepossibility that a junction of metal and FE or NB (as op-posed to bulk materials or to a junction metal/SC) pro-duces high thermopower. This is possible if a δ-layer ofsuitable rare-earth impurity atoms substitutes the origi-nal layer at the interface. The localized character of theimpurity f -orbital has a strong effect on the transmissionof carriers across the junction. The figure of merit ZTattained is very high, especially at low temperatures. Arealistic device is proposed which exploits the thermo-electric potentialities of the junction.

Acknowledgments

This work is supported by the NSF contract DMR0099572, MIUR-FIRB “Quantum phases of ultra-lowelectron density semiconductor heterostructures”, andMIUR Progetto Giovani Ricercatori. The authors thank

16

Professors D. Chemla, M. L. Cohen and S. G. Louie forthe hospitality of University of California Berkeley, wherethis work was initiated. M. R. is grateful to ProfessorsFranca Manghi and Elisa Molinari for all the help given.L. J. S. thanks J. E. Hirsch for helpful discussions andthe Miller Institute for support. Con il contributo delMinistero degli Affari Esteri, Direzione Generale per laPromozione e la Cooperazione Culturale.

APPENDIX A: CURRENT OPERATOR

In this appendix we derive the form of the densitycurrent operator J for the effective Hamiltonian HMF

of Eq. (18) (or HMF −µ). This is a key quantity in com-puting the transport properties of the junction.One way to find J is to exploit the gauge invariance of

the hamiltonian (18). In the presence of an oscillating,uniform electric field E(t) = Ee−iωt along z (t is the timeand ω/2π is the frequency), one must add to HMF a termrepresenting the coupling to the applied field. One thusintroduces the vector potential A(t) = cEe−iωt/iω such

that E(t) = −A/c, where c is the speed of light. Thegauge transformation

d′j = dj exp

[

− ie

hcA(t) aj

]

,

f ′j+1/2 = fj+1/2 exp

[

− ie

hcA(t) a (j + 1/2)

]

, (A1)

removes the coupling to the vector potential A(t) pre-serving the gauge invariance of HMF; if we apply thetransformation (A1) to HMF we obtain a new Hamilto-

nian HMF′ which must be given, expanding to first orderin A, by

HMF′ ∼ HMF − aJ

cA. (A2)

Equation (A2) permits to identify J , whose expression is

J =∑

j

J(j) , (A3)

J(j) =i

het′ d†j+1dj + H.c. ∀j < 0, (A4)

J(0) =i

het d†1d0 +

ie

2h

[ (

∆L + VdfL

)

f †1/2d0

−(

∆R − VdfR

)

f †1/2d1

]

+ H.c., (A5)

J(j) =i

het d†j+1dj +

ie

2h

[

(∆ + Vdf ) f†

j+1/2dj

− (∆− Vdf ) f†

j+1/2dj+1

]

+ H.c. ∀j > 0. (A6)

Here J(j) is the electric current operator at atomic site j(in one dimension the current coincides with its density).If the transport across the junction is stationary, at eachsite j the current must be conserved; thus, to identifythe density current associated with each quasi-particleexcitation, we can calculate the operator J(j) at somesuitable atomic site j where its expression is simpler: thisis used in Sec. III. For j < 0 HMF is a simple tight-binding Hamiltonian, and the formula for J(j) becomes,replacing d and f operators in (A4) with γ operators:

J(j) =2et′

hNs

∑

k

Im [u∗k(j)uk(j + 1)] γ†keγke

− 2et′

hNs

∑

k

Im[

u−k(j) u∗−k(j + 1)

]

γ†−khγ−kh

+2et′

hNs

∑

k

Im[

u−k(j) u∗−k(j + 1)

]

. (A7)

It turns out that the constant third term on the right-hand side of Eq. (A7) is always zero, hence it is easy toidentify the electric current Jke(j) and J−kh(j) associ-ated with each electron or hole excitation, respectively:

J(j) =1

Ns

∑

k

[

Jke(j) γ†keγke

+ J−kh(j) γ†−khγ−kh

]

, (A8)

Jke(j) =2et′

hIm [u∗k(j)uk(j + 1)] ∀j < 0, (A9)

J−kh(j) = −2et′

hIm

[

u−k(j)

× u∗−k(j + 1)]

∀j < 0. (A10)

The particle current density for electrons JkeN (j) is givenby

JkeN (j) =Jke(j)

e, (A11)

and for holes J−khN (j) by

J−khN (j) =J−kh(j)

−e. (A12)

The picture consistent with these results is that(uk(j) , vk(j + 1/2)) [or (u, v)] represents the two-component electron (hole) wavefunction of the elemen-tary excitation, and JkeN (j) (J−khN ) is its probabilitycurrent density. Indeed, equations (A9-A10) can be de-rived also from the continuity equation of wavefunctions,if one associates the electron amplitude (u, v) with thecharge e, and the hole amplitude (u, v) with −e. Toobtain the continuity equation, one has to multiply thetime-dependent BdG equation (46) [(47)] by u∗k(j, t) [u

∗],

17

and the complex conjugate of (46) [(47)] by uk(j, t) [u],and finally subtract one term from the other.In an analogous way we can also write the expression of

the current for j > 0. If ∆ and Vdf are real, one obtains:

Jke(j) =2et

hIm [u∗k(j) uk(j + 1)] +

e

h(∆ + Vdf )

× Im [u∗k(j) vk(j + 1/2)]− e

h(∆− Vdf )

× Im [u∗k(j + 1) vk(j + 1/2)] , (A13)

J−kh(j) = −2et

hIm

[

u−k(j) u∗−k(j + 1)

]

− e

h(∆ + Vdf )

× Im[

u−k(j) v∗−k(j + 1/2)

]

+e

h(∆− Vdf )

× Im[

u−k(j + 1) v∗−k(j + 1/2)]

. (A14)

18

∗ Electronic address: rontani@unimore.it;URL: http://www.nanoscience.unimo.it/max_index.html

1 Gerald Mahan, Brian Sales and Jeff Sharp, Phys. Today50 (3), 42 (1997). For a review see G. D. Mahan, “GoodThermoelectrics”, in Solid State Physics 51, ed. H. Ehren-reich and F. Spaepen, Academic Press, (1998), p. 81.

2 A value of ZT = 2.4 at room temperature has been re-cently reported for p-type Bi2Te3/Sb2Te3 superlattice de-vices, see R. Venkatasubramanian, E. Siivola, T. Colpitts,and B. O’Quinn, Nature (London) 413, 597 (2001).

3 A preliminary report of the results of this work was givenin Massimo Rontani and L. J. Sham, Appl. Phys. Lett. 77,3033 (2000).

4 See B. C. Sales, D. Mandrus, and R. K. Williams, Science272, 1325 (1996), and references therein.

5 G. D. Mahan and J. O. Sofo, Proc. Natl. Acad. Sci. USA,93, 7436 (1996).

6 G. D. Mahan, J. Appl. Phys. 65, 1578 (1989); J. O. Sofoand G. D. Mahan, Phys. Rev. B 49, 4565 (1994).

7 Wenjin Mao and Kevin S. Bedell, Phys. Rev. B 59, R15590(1999).

8 L. D. Hicks and M. S. Dresselhaus, Phys. Rev. B 47, 12727(1993); L. D. Hicks, T. C. Harman, and M. S. Dresselhaus,Appl. Phys. Lett. 63, 3230 (1993).

9 J. O. Sofo and G. D. Mahan, Appl. Phys. Lett. 65, 2690(1994).

10 L. D. Hicks, T. C. Harman, X. Sun, and M. S. Dressel-haus, Phys. Rev. B 53, R10493 (1996); T. C. Harman,D. L. Spears, and M. P. Walsh, J. Electronic Materials,28, L1 (1999).

11 T. Yao, Appl. Phys. Lett. 51, 1798 (1987);S. M. Lee, D. G. Cahill, and R. Ventakasubramanian,Appl. Phys. Lett. 70, 2957 (1997); G. Chen and M. Neagu,Appl. Phys. Lett. 71, 2761 (1997); P. Hyldgaard andG. D. Mahan, Phys. Rev. B 56, 10754 (1997); G. Chen,Phys. Rev. B 57, 14958 (1998); M. V. Simkin andG. D. Mahan, Phys. Rev. Lett. 84, 927 (2000).

12 M. Nahum, T. M. Eiles, and John M. Martinis,Appl. Phys. Lett. 65, 3123 (1994).

13 H. L. Edwards, Q. Niu, G. A. Georgakis, and A. L. deLozanne, Phys. Rev. B 52, 5714 (1995).

14 G. D. Mahan and L. M. Woods, Phys. Rev. Lett. 80, 4016(1998); G. D. Mahan, J. O. Sofo, and M. Bartkowiak,J. Appl. Phys. 83, 4683 (1998).

15 B. Moyzhes and V. Nemchinsky, Appl. Phys. Lett. 73,

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16373 (1993)]. In our case [GT ]=W/K, in the former case[GT ]=W/(K m2).

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